The effect of network topology on optimal exploration strategies and the evolution of cooperation in a mobile population

We model a mobile population interacting over an underlying spatial structure using a Markov movement model. Interactions take the form of public goods games, and can feature an arbitrary group size. Individuals choose strategically to remain at their current location or to move to a neighbouring location, depending upon their exploration strategy and the current composition of their group. This builds upon previous work where the underlying structure was a complete graph (i.e. there was effectively no structure). Here, we consider alternative network structures and a wider variety of, mainly larger, populations. Previously, we had found when cooperation could evolve, depending upon the values of a range of population parameters. In our current work, we see that the complete graph considered before promotes stability, with populations of cooperators or defectors being relatively hard to replace. By contrast, the star graph promotes instability, and often neither type of population can resist replacement. We discuss potential reasons for this in terms of network topology.


S 1: I
In this scenario, we vary the population size, exploration time, and the rewardto-cost ratio. We present the results for each of the three network structures (complete, circle, and star graphs), and within each structure we consider the e ects of the population size, exploration time, and the reward-to-cost ratio in turn, while all other parameters assume their base values.
1.1. Complete graph. In gure 1 we see the optimal staying propensities and xation probabilities for the complete graph over the full range of population sizes that we considered (i.e., 10-50). The staying propensities behave in essentially the same way for all population sizes. Resident defectors use (as in all cases) the maximum value of 0.99, with mutant cooperators increasing from a high level to the maximum as the movement cost increases. Resident cooperators vary between a staying propensity of 0.2 for low cost to one of 0.6 for a high cost; mutant defectors increase from the minimum value of 0.01 to the maximum of 0.99, with intermediate values over a small range of costs only. In all cases selection favours cooperators for movement costs up to 0.2. Selection favours defectors only in su ciently small populations and for su ciently large movement costs. The le -column panels show optimal staying propensities for resident defectors and mutant cooperators, the middle-column panels show optimal staying propensities for resident cooperators and mutant defectors, and the right-column panels show the fixation probabilities of best mutant cooperators and best mutant defectors in the resident defector and cooperator populations, respectively. Legend: defectors-red with star markers for data points, cooperators-blue with round markers for data points; residents-solid line, mutants-dashed line.
Changing exploration time has little e ect on the staying propensities. It is clear that longer exploration times are favourable to cooperators, who have more time to nd and bene t from other cooperators. In particular, shortening the exploration time as in gure 2 means selection favours defectors for suciently high movement cost even in large populations, whilst lengthening the exploration time as in gure 3 results in selection favouring cooperators up to a higher movement cost threshold. In gures 4 (population size 10) and 5 (population size 50) we consider varying the reward-to-cost ratio from 1 to 50, in each case considering movement costs 0.1, 0.5, and 0.9 only. In all cases (except for resident defectors), staying propensity decreases over the range of reward-to-cost ratios because receiving larger reward of cooperation partially o sets potential movement costs. As we would expect, cooperators do better relative to defectors when the ratios are high. In particular in both gures we see that selection favours cooperators for su ciently high ratios, and selection favours defectors for su ciently low ratios, with a range of intermediate values (of increasing width with movement cost) where selection opposes change.  1.2. Circle graph. In gure 6 we consider the optimal staying propensities and xation probabilities over the full range of population sizes, as we did for the complete graph in gure 1. The staying propensities follow the pattern similar to the complete graph case. There is generally precisely one dominant interactive strategy (except in the case for = 10), as we see from the xation probability lines crossing when both achieve the neutral value. Selection favours cooperators for movement costs below a threshold ( = 0.4), and selection favours defectors for movement costs above the threshold. In gures 7 (exploration time = 5) and 8 (exploration time = 25) we investigate changes in the exploration time for the circle graph. The staying propensities again behave as for the complete graph. Usually there is one dominant strategy: cooperators for longer exploration time and defectors for shorter exploration time. In gures 9 and 10 we vary the reward-to-cost ratio for population sizes = 10 and = 50 for movement costs 0.1, 0.5, and 0.9. We again observe that generally one strategy is dominant: cooperators if the reward-to-cost ratio is su ciently large, and defectors if the reward-to-cost ratio is su ciently small. 1.3. Star graph. In gure 11 we consider optimal staying propensities and xation probabilities for various population sizes on our third structure, the star graph. Mutant cooperators have much lower staying propensity than on the complete and circle graphs. In the hub-and-spoke topology of the star graph, a cluster of cooperators is most likely to form in the hub (the centre of the star), and hence it is bene cial to get from the leaves to the centre quickly. For this same reason, resident cooperators evolve to the lowest staying propensity ( = 0.01) for all movement costs and population sizes. Mutant defectors do best by imitating the strategy "get to the centre quickly", and they prefer to use low staying propensities most of the time. The xation probabilities for cooperators follow a similar pattern to the other graphs, where there is a movement cost threshold below which the xation probability of cooperators exceeds the neutral drift one; this threshold increases with the population size. The most striking feature of gure 11, however, is that for all combinations of population size and movement cost, the xation probability of mutant defectors always exceeds the neutral drift one. It is easy for defectors to locate and exploit the cooperating cluster in the centre of the star.
In gures 12 and 13 we consider shorter and longer exploration times for population sizes 10 and 50. There is not a large e ect on the staying propensities, except that the mutant defectors choose a higher staying propensity than before when the exploration time is longer, in gure 13. Interestingly both shortening the exploration time and extending the exploration time makes it more di cult for mutant cooperators to replace resident defectors. Thus intermediate values of the searching time seem optimal for cooperator invaders. On the one hand, cooperators need to spend su cient time together to accumulate the rewards; hence shortening the exploration time hurts cooperators. On the other hand, in the hub-and-spoke topology of the star graph, slow-moving resident defectors (with staying propensity = 0.99) are more likely to eventually move to the centre and thus exploit a potential cooperating cluster there. So, too long an exploration phase also hurts cooperators. In gures 14 and 15 we vary the reward-to-cost ratio for the star graph with population sizes 10 and 50 for movement costs 0.1, 0.5, and 0.9. The equilibrium staying propensity of resident cooperators increases to the maximum value 0.99 when the reward-to-cost ratio is smallest ( / = 1) and the movement cost is su ciently high. In this case, the low value of the reward of cooperation does not compensate the potentially high cost of movement. As before, the xation probability of mutant defectors always exceeds the neutral drift one. Mutant cooperators can replace resident defectors for su ciently high reward-to-cost ratio for any movement cost.

S 2: I
In this scenario, we only consider two populations sizes (10 and 50), and we vary the exploration time and the reward-to-cost ratio. Similarly to the scenario 1, we group the results by the network structure.
2.1. Complete graph. Figure 16 shows the equilibrium staying propensities for cooperators and defectors in a mixed (half-and-half) population with normal exploration time ( = 10), the xation probabilities of cooperators and defectors using the equilibrium staying propensities in a mixed population, and the xation probabilities of cooperator and defector mutants in a defector and cooperator resident population, respectively, where both mutants and residents inherit staying propensities from the mixed population. Defectors evolve their staying propensity to the lowest possible one (0.01) for low movement costs and the highest possible one (0.99) for high movement costs; they rarely utilize intermediate staying propensities. In contrast, cooperators never evolve to the extreme staying propensities; their equilibrium staying propensity increases from 0.2 to 0.9 with the movement cost. The le -column panels show mutually best staying propensities for cooperators and defectors in a mixed population, the middle-column panels show fixation probabilities of cooperators and defectors with mutually best staying propensities in a mixed population, and the right-column panels show the fixation probabilities of mutant cooperators and mutant defectors in a resident defector and cooperator populations, respectively, where both cooperators and defectors inherit staying propensities from the mixed population. Legend: defectors-red with star markers for data points, cooperators-blue with round markers for data points; mixed population-solid line, mutants-dashed line.
In a small mixed population, cooperators do better than defectors for intermediate movement costs (0.1-0.6); defectors prevail otherwise. In a large mixed population, cooperators dominate defectors for all movement costs except the highest one (0.9). The mutant-residents population shares the basic qualitative behavior with scenario 1: selection usually opposes change. In a small mutant-residents population, selection opposes change for all positive movement costs. In a large mutant-residents population, selection favours cooperators for movement costs 0.1-0.3 and opposes change otherwise. In general, cooperators do better in larger populations on the complete graph. Figures 17 and 18 show the e ect of the exploration time. Shortening the exploration time signi cantly hurts cooperators and extending the exploration time signi cantly helps cooperators in both mixed and mutant-residents populations.
With a short exploration phase ( = 5), defectors do better than cooperators in a small mixed population for all movement costs except 0.2, and defectors dominate cooperators in a large mixed population for all movement costs above 0.3. In a mutant-residents population, selection favours defectors for su ciently high movement cost and for zero movement cost in a small population. The xation probability of mutant cooperators never exceeds the neutral drift one if the exploration time is too short. Figures 19 and 20 explore the e ect of varying the reward-to-cost ratio for movement costs 0.1, 0.5, and 0.9.
If the reward-to-cost ratio is too small then defectors dominate cooperators in all population types, and if the ratio is su ciently large then cooperators dominate defectors in a mixed population. Interestingly, selection never favours cooperators in a mutant-residents small population for low movement cost ( = 0.1). In this case, resident defectors inherit very low staying propensity from the mixed population, and hence they are better suited to resist mutants in a low-movement-cost environment.  Figure 21 presents the base case for the circle graph. Similarly to the complete graph, defectors rarely evolve to intermediate staying propensities in a mixed population, and cooperators do the opposite: they rarely use extreme staying propensities. In a mixed population, cooperators do better then defectors for intermediate movement costs 0.1-0.4. Defectors do better than cooperators for all other movement costs except in a large population where they are tied with cooperators for movement costs 0 and 0.5. In a large mutant-residents population, the situation is similar to the mixed population case: selection favours cooperators for intermediate movement costs, selection favours defectors for high movement costs, and selection opposes change at two threshold movement cost values 0 and 0.5. In a small mutantresidents population, selection favours cooperators only for movement cost 0.2. Selection favours defectors for zero and high movement costs, and selection opposes change otherwise. 2.3. Star graph. The star graph presented a unique challenge for the mixed population case: there were no equilibrium staying propensities for cooperators and defectors in a large mixed population for movement costs above 0.1. See the main text for an explanation and our proposed resolution. Figure 26 presents the base case results. In a mixed population, cooperators do slightly better than defectors for a range of intermediate movement costs; defectors do better than cooperators for low and high movement costs. In a mutant-residents population, the xation probability of cooperators never exceeds the neutral drift one. Selection favours defectors for most values of the movement cost. Overall, defectors do better than cooperators on the star graph even in the non-rare interactive mutations case. Figures 27 and 28 investigate various exploration times. Shortening the exploration phase to = 5 results in complete dominance of defectors for all other parameter combinations. Extending the exploration phase to = 25 nally allows cooperators to dominate defectors in a mixed population over all but lowest movement costs. In a mutant-residents population, mutant defectors can replace residents cooperators only for movement costs 0-0.2, and mutant cooperators can replace resident defectors for movement costs 0.1-0.6 in a large population. Figures 29 and 30 summarise the outcomes for varying the reward-to-cost ratio. This ratio has no e ect in small populations and low movement cost (0.1): defectors dominate cooperators in both mixed and mutant-residents populations. Otherwise, the outcomes for a mixed population aligns with other structures: defectors do better for small values of the reward-to-cost ratio, and cooperators do better for large values of the reward-to-cost ratio. Similar situation is observed in a mutant-resident population except in a small population for intermediate movement cost (0.5), where selection favours change for large values of the reward-to-cost ratio. In this parameter regime, defectors evolve to su ciently low staying propensity (cf. gure 29d), while cooperators do not. Also, the xation probability of mutant defectors always exceeds the neutral drift one in a large population and low movement cost (0.1) because defectors evolve to lower staying propensities than cooperators for high reward-to-cost ratio in a mixed population (cf. gure 30a).